CVaR Distortion Function Derivation

Article Quant Q&A · Author: T-at-R

Summary

The document discusses how to derive a distortion function for conditional value at risk (CVaR), defined through an integral of the generalized inverse of a loss distribution. It identifies the proposed piecewise function and explains that the derivation relies on connecting the integrated survival function with the quantile integral and expected value.

The accepted answer extends that relationship to variables that may take negative values, expressing the quantile integral through survival probabilities on both sides of zero. It then points to changing variables in indicator functions and handling separate cases according to whether the survival probability is above or below the CVaR threshold. A second response suggests starting directly from the quantile definition. The discussion provides a proof outline rather than a complete worked derivation, and the two responses use different tail-probability conventions, so readers should check how the confidence level and tail are defined before applying the formula.

Key ideas

The quantile integral can be related to an integral of survival probabilities.
For variables that can be negative, the survival-integral expression needs a correction below zero.
The CVaR distortion derivation can be organized by splitting into cases around the tail threshold.
The quantile definition offers an alternative starting point for deriving the distortion function.

Tags

Statistics Risk management

Full text

# How to calculate the distortion function for CVaR?


# How to calculate the distortion function for CVaR?












Can anyone give me some hints as to how to prove that

$$g(x) = \begin{cases} \frac{x}{1-\alpha}, &0 \leq x \leq 1-\alpha\\ 1 , &1-\alpha \leq x \leq 1 \end{cases}$$

Is the distortion function corresponding to $\text{CVaR}_\alpha(X)$?

Here I define $$\text{CVaR}_\alpha(X) = \frac{1}{\alpha} \int_{0}^{\alpha} F_X^{-1}(u) du$$ For more details, see Expected shortfall - Wikipedia .Of course the inverse is supposed to be understood as the generalized inverse.

My problem is that a direct calculation does not seem to work for me. Maybe I am missing some trick.

Any help would be appreciated.

## Answer by T-at-R (score 2, accepted)

https://quant.stackexchange.com/a/31101

I have solved it myself. The key was to realize that for $X \geq 0$ and $S_X(t) = \mathbb{P}(X>t)$

$$ \int_0^\infty S(t) dt = \int_0^1 F_X^{-1}(u) du = \mathbb{E}\left[X \right].$$

This is elegantly explained in Characterization of $\mathbb{E}$.

Now this relationship can be extended for the whole real line, thus

$$ \int_0^1 F_X^{-1}(u) du = \int_0^\infty S_X(t) dt + \int_{-\infty}^0 S_X(t) -1 dt$$.

The rest of the proof is a matter of changing the variables in the indicator functions and considering the two cases of (a) $S_X(t) \geq 1-\alpha$ and (b) $S_X(t) \leq 1-\alpha$.

For a direct calculation @CaffeRistretto tipp was very helpful. So it is best to start with the definition of CVaR and work towards the distortion function.

## Answer by CaffeRistretto (score 1)

https://quant.stackexchange.com/a/31070

Maybe prove that

$$CVaR_\alpha (X) = \frac{1}{\alpha} \int_0^\alpha F^{-1}_X(u) du$$

has the distortion function

$$ g(u)= \begin{cases} \frac{u}{\alpha}, \quad \; u \leq \alpha \\ 1, \qquad u > \alpha\end{cases}$$

would be easier?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Deriving the Distortion Function for Conditional Value at Risk

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